Even Function

Q. If the real valued function $f\left(x\right)=\frac{a^x-1}{x^n(a^x+1)}$ is even, then $n$ is equal to

1. $2$
2. $\frac{2}{3}$
3. $\frac{1}{4}$
4. $3$

Q. Let $f\left(x\right)=\left(\right[a{]}^2-5\left[a\right]+4)x^3-(6\{a{\}}^2-5\{a\}+1)x-\left(\tan x\right)\times sgnx$ be an even function for all $x\in R$, then the sum of all possible values of ${}^{\prime }3a^{\prime }$
is
(where, [.] and {.} denote the greatest integer function and fractional part functions, respectively)

Q. Which among the following is/are odd functions?

1. $f\left(x\right)=x\left(\frac{e^x-1}{e^x+1}\right)$
2. $g\left(x\right)=\frac{a^x-1}{a^x+1}$
3. $h\left(x\right)=\log (x^6+\sqrt{x^2+1})$
4. $k\left(x\right)=x(x^2-|x|)$

Q. If $f(x+y)=f\left(x\right)\cdot f\left(y\right)$ for all $x,y\in R$ and $f\left(0\right)\ne 0$, then the function $g\left(x\right)=\frac{f\left(x\right)}{1+\left\{f\right(x){\}}^2}$ is

1. an odd function
2. an even function
3. Neither an even nor an odd function
4. an odd function for $x<0$

Q. If in the equation y = f(x), f(x) is even function then

1. $f(-x)=-f\left(x\right)$
2. $f(-x)=f\left(x\right)$
3. Its graph is symmetric about y-axis.
4. If (x, y) is a point on its graph, then (-x, y) is also on the same graph.

Q. If $y=x\left\{x\right\}-x\left[x\right],$ then $\frac{dy}{dx}$ is equal to
$($Where $x\ne Z,[\cdot ]$ denotes the greatest integer function and $\{\cdot \}$ denotes the fractional part function$)$

1. $\left[x\right]$
2. $x-1$
3. $\left\{x\right\}$
4. $2\left\{x\right\}$

Q. Let $f\left(x\right)=\left(\right[a{]}^2-5\left[a\right]+4)x^3+(6\{a{\}}^2-5\{a\}+1)x-\left(\tan x\right)\times \text{sgn}x$ be an even function for all $x\in R,$ then the sum of all possible values of $a$ is (where $[.],\{.\},\text{sgn}x$ represents greatest integer function , fractional part function and signum function respectively)

1. $\frac{17}{6}$
2. $\frac{53}{6}$
3. $\frac{31}{3}$
4. $\frac{35}{3}$

Q. Let $f\left(x\right)=a^x(a>0)$ be written as $f\left(x\right)=f_1\left(x\right)+f_2\left(x\right)$, where $f_1\left(x\right)$ is an even function and $f_2\left(x\right)$ is an odd function. Then $f_1(x+y)+f_1(x-y)$ equals :

1. $2f_1\left(x\right)f_1\left(y\right)$
2. $2f_1(x+y)f_1(x-y)$
3. $2f_1\left(x\right)f_2\left(y\right)$
4. $2f_1(x+y)f_2(x-y)$

Q. The function $f:R\to [-\frac{1}{2},\frac{1}{2}]$ defined as $f\left(x\right)=\frac{x}{1+x^2},$ is

1. neither injective nor surjective
2. surjective but not injective
3. injective but not surjective
4. invertible

Q.

Let A={-1, 0, 1, 2}, B={-4, -2, 0, 2} and $f,g:A\to B$ be the function defined by $f\left(x\right)=x^2-x,x\in A$ and $g\left(x\right)=2|x-\frac{1}{2}|-1,x\in A.$
Are f and g equal ? Justify your answer.

Q. Graph of 1. |y| = |1-|x-1|| 2. |y| = ({x} - 1)² where { } represents fractional part 3. |y| = |2-1/(|x-1|)| 4. [|Y|] = 4- [|x|] where [ ] represents GIF

Q. The term independent of $x$ in ${\left[\sqrt{\frac{x}{3}}+\sqrt{\left(\frac{3}{2x^2}\right)}\right]}^{10}$ is

1. $1$
2. $\frac{5}{12}$
3. ${}^{10}{C}_1$
4. none

Q. Let $f\left(x\right)=\left(\right[a{]}^2-5\left[a\right]+4)x^3-(6\{a{\}}^2-5\{a\}+1)x-\left(\tan x\right)\times sgnx$ be an even function for all $x\in R$, then the sum of all possible values of ${}^{\prime }3a^{\prime }$
is
(where, [.] and {.} denote the greatest integer function and fractional part functions, respectively)

Q. Evaluate : $\int \frac{(x^2+1)(x^2+4)}{(x^2+3)(x^2-5)}dx$

Q. if $f:R\to R,f\left(x\right)=2x+1$ and $g:R\to R,g\left(x\right)=x^3$, then $\left(gof{)}^{-1}\right(27)$ equals-

1. $-1$
2. $0$
3. $1$
4. $2$

Q. Let $f$ be a function defined by $f\left(x\right)=\frac{x-5}{x-3};x\ne 3$, $f^k\left(x\right)$ denote the composition of $f$ with itself taken $k$ times i.e $f^3\left(x\right)=f\left(f\right(f\left(x\right)\left)\right)$
Then

1. $f^{2019}\left(2020\right)=\frac{6055}{2019}$
2. $f^{2020}\left(2021\right)=2021$
3. $f^{2022}\left(2019\right)=\frac{4033}{2017}$
4. $f^{2021}\left(2020\right)=\frac{2015}{2019}$

Q. If in the equation y = f(x), f(x) is even function then

1. $f(-x)=-f\left(x\right)$
2. $f(-x)=f\left(x\right)$
3. Its graph is symmetric about y-axis.
4. If (x, y) is a point on its graph, then (-x, y) is also on the same graph.

Q. If $f:R\to R,f\left(x\right)=2x-3;g:R\to R,g\left(x\right)=x^3+5$, then the value of $\left(fog{)}^{-1}\right(x)$:

1. ${(x-\frac{7}{2})}^{1/3}$
2. ${\left(\frac{x-2}{7}\right)}^{1/3}$
3. ${\left(\frac{x+7}{2}\right)}^{1/3}$
4. ${\left(\frac{x-7}{2}\right)}^{1/3}$

Q. Consider the function
$f\left(x\right)=-2x^3-9x^2-12x+1$
The function $f\left(x\right)$ is an increasing function in the interval

1. $(-2,-1)$
2. $(-\infty ,-2)$
3. $(-1,2)$
4. $(-1,\infty )$

Q. The function $f\left(x\right)=\frac{1}{x^4+x^2+1}$ is

1. Discontinuous at $x=0$
2. Discontinuous at $x=1$
3. Discontinuous at $x=-1$
4. Continuous at every where

Q.

If f is an even function defined on the interval (-5, 5), then the real values of x, satisfying the equation $f\left(x\right)=f\left(\frac{x+1}{x+2}\right)$ are ___.

1. $\frac{3\pm \sqrt{5}}{2}$

2. $\frac{5\pm \sqrt{3}}{2}$

3. $\frac{-3\pm \sqrt{5}}{2}$

4. $\frac{-5\pm \sqrt{3}}{2}$

Q. Suppose $y=f\left(x\right)$ is the solution of the differential equation $(1-x^2)\frac{dy}{dx}=\left(\sqrt{1-x^2}\right)(x^4+2x)+xy$ for $x\in (a,b)$, where $f\left(0\right)=0$. Then

1. $f\left(x\right)$ is an even function.
2. $f\left(x\right)$ is an odd function.
3. $b-a=2$
4. $\underset{-1/2}{\overset{1/2}{\int }}f\left(x\right)dx=\frac{\pi }{6}-\frac{\sqrt{3}}{4}$

Q. If $f$ is even or odd function then

Q.

If f is an even function defined on the interval (-5, 5), then the real values of x, satisfying the equation $f\left(x\right)=f\left(\frac{x+1}{x+2}\right)$ are ___.

1. $\frac{3\pm \sqrt{5}}{2}$

2. $\frac{5\pm \sqrt{3}}{2}$

3. $\frac{-3\pm \sqrt{5}}{2}$

4. $\frac{-5\pm \sqrt{3}}{2}$

Q. Sketch the graph of Y = -{x}

Q. Find the equation of the curve that passes through the point (0, a) and is such that at any point (x, y) on it, the product of its slope and the ordinate is equal to the abscissa.

Q. Let $f(x+y)=f(x.f(y\left)\right)\forall x,y\in R$ and $y\in R$ and $f\left(0\right)\ne 0$. Then the function $\varphi$ defined by $\varphi \left(X\right)=\frac{f\left(x\right)}{1+\left(f\right(x){)}^2}$ is

1. Neither odd nor even function
2. An odd function
3. Constant function
4. An even function

Q. If $\sum _{r=1}^{n}r(r+1)(2r+3)=an+b{n}^3+c{n}^2+dn+e$, then

1. $c=9/2$
2. $a=1/2$
3. $b=8/3$
4. $e=0$

Q. If $(x-1{)}^4-16=0$, then the sum of nonreal complex values of x is

1. 2
2. 0
3. 4
4. none of these

Q.

If f is an even function defined on the interval (-5, 5), then the real values of x, satisfying the equation $f\left(x\right)=f\left(\frac{x+1}{x+2}\right)$ are ___.

1. $\frac{3\pm \sqrt{5}}{2}$

2. $\frac{5\pm \sqrt{3}}{2}$

3. $\frac{-3\pm \sqrt{5}}{2}$

4. $\frac{-5\pm \sqrt{3}}{2}$